Local behaviour of first passage probabilities

By R. Doney. 2010

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Suppose that S is an asymptotically stable random walk with norming sequence cn and that Tx is the time that S first enters (x, ∞), where x ≥ 0. The asymptotic behaviour of P(T0 = n) has been described in a recent paper of Vatutin and Wachtel (Probab. Theory Relat. Fields 143:177–217, 2009), and here we build on that result to give three estimates for P(Tx = n), which hold uniformly as n → ∞ in the regions x = o(cn), x = O(cn), and x/cn → ∞, respectively.

Subfields: Artificial Intelligence, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e873d2f706376067be94bce96c464e13a06d9577